The Mirror reports that Astori was on the shopping list before Dejan Lovren joined the Saints in January.
The 26-year-old has also been strongly linked with the likes of AC Milan and Everton.
The £10million-rated defender played for Italy in the Confederations Cup in Brazil last summer.
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Being an amiable guy is good for you and Jurgen Klopp is living proof.
Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.